When You Come In…

Sit at tables with other 4th or 5th grade teachers, not necessarily from your own school.Did you bring samples of student work on the assessment?

3-4-5 Continuum

Multiplication and division follow a learning progression across the grades.

Starting Point

One End Goal

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Middle School

High School

P.O.P. for this session

• Purpose – Develop professional knowledge and effective teaching strategies

• Outcomes – Responsive teaching, leading to greater readiness for the next grade

• Processes – Collaborative discussion, videos, problem types, learning progressions, diagnostic assessments, resources, virtual manipulatives

Professional Knowledge

• Textbooks like Everyday Math and EnVision are important instructional resources, but:

• Professional knowledge is essential to know where each child is on the learning progression and how to help them move to the next step

• Professional knowledge includes both deep content knowledge and a broad repertoire of teaching strategies

Professional Knowledge

• Content knowledge: CCSS, Learning Progressions, Common Error Patterns

• Instructional strategies: IES Practice Guide, Concrete-Representational-Abstract (CRA)

Key Questions

1) What must a student know in order to know multiplication (and division)?

2) How is multiplication of decimals like multiplication of whole numbers?

3) How is multiplication of fractions like multiplication of whole numbers?

4) How do we help all students become successful with multiplication?

Common Core for 4th grade

Critical Area 1: Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends;

• Use the four operations with whole numbers to solve problems.

• Generalize place value understanding for multi-digit whole numbers.

• Use place value understanding and properties of operations to perform multi-digit arithmetic. (SBAC)

Common Core for 4th grade

Critical Area 2: Developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers;

• Extend understanding of fraction equivalence and ordering. • Build fractions from unit fractions by applying and extending

previous understandings of operations on whole numbers. • Understand decimal notation for fractions, and compare

decimal fractions. (SBAC)Critical Area 3: Understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.

Common Core for 5th Grade

Critical Area 1: Extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations;

• Understand the place value system. • Perform operations with multi-digit whole numbers and

with decimals to hundredths. (SBAC)

Common Core for 5th Grade

Critical Area 2: Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions);

• Use equivalent fractions as a strategy to add and subtract fractions. • Apply and extend previous understandings of multiplication and

division to multiply and divide fractions. (SBAC)

Critical Area 3: developing understanding of volume.• Geometric measurement: understand concepts of volume and

relate volume to multiplication and to addition. (SBAC)

Common Core Standards

• Using the Common Core Standards for 3rd-5th grade, make a map of the learning progression for multiplication and division.

• Decide among yourselves what the “end goal” is.

Key Instructional Strategies

• Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

• Interventions should include instruction on solving word problems that is based on common underlying structures.

• Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas.

Key Instructional Strategies

• Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.

• Include motivational strategies in tier 2 and tier 3 interventions.

Explicit Instruction

• Teachers provide clear models for solving a problem type using an array of examples.

• Students receive extensive practice in use of newly learned strategies and skills.

• Students are provided with opportunities to think aloud (i.e., talk through the decisions they make and the steps they take).

• Students are provided with extensive feedback.National Mathematics Advisory Panel, 2008

Explicit Instruction

• Ensure that instructional materials are systematic and explicit. In particular, they should include numerous clear models of easy and difficult problems, with accompanying teacher think-alouds.

• Provide students with opportunities to solve problems in a group and communicate problem-solving strategies.

• Ensure that instructional materials include cumulative review in each session.

Explicit Instruction

The NMAP notes that this does not mean that all mathematics instruction should be explicit.So what’s the difference?1) Do you see a pattern? (3.OA.9)2) Can you make a connection?3) Can you solve problems independently?

When?

• You need to paint the walls of the cafetorium in your school. You have to figure out how much paint is needed. The walls of the room are 24 feet high. You measure each wall along the bottom, and get these measurements: Wall 1: 53 feet long. Wall 2: 34 feet long, Wall 3: 53 feet long, Wall 4: 34 feet long. There are also two doors in the gym, each is 10 feet high by 8 feet wide, and you don’t have to paint them. How many square feet of paint would you need for one coat on all the walls?

Underlying structure

• Teach students about the structure of various problem types, how to categorize problems based on structure, and how to determine appropriate solutions for each problem type.

• Teach students to recognize the common underlying structure between familiar and unfamiliar problems and to transfer known solution methods from familiar to unfamiliar problems.

Visual Representations

The C-R-A Approach

• Moving from concrete to representational to abstract• … from objects to pictures to symbols

Braided Knowledge

Knowledge of the content is interwoven with knowledge of effective teaching strategies to produce deep learning.

Content

• CCSS• Learning Progressions• Common Error Patterns

Basic concepts

1. Concepts of multiplication and division need to be understood through problem situations.

2. Fact family relationships are developed by solving many problems. Students often develop strategies for quickly producing a fact.

3. Fluency comes gradually. 4. Number sentences should always be written when solving

problems – even if they can be solved by alternative methods (e.g. repeated adding) – to connect the symbols to the operations. Some problems involve “missing factors” such as 5 people will ride in each car. How many cars do we need for 35 people? 5 x ___ = 35.

Basic problem types

Multiplication is1. Equal groups of… (3.OA.1)2. So many times larger (or smaller)… (4.OA.1)

Basic problem types

Multiplication is: Equal groups of… • Megan has 5 bags of cookies. There are 3 cookies in each bag.

How many cookies does Megan have all together?• Rate problems: I eat 3 bananas a day. How many have I eaten

after 5 days?• Price problems: One piece of candy costs 5 cents. How much

does 8 pieces cost?• Array problems: There are 4 rows of desks in a classroom with

8 desks in each row. How many desks are there altogether?• Combination problems: At the ice cream store they sell 7

flavors of ice cream and 3 kinds of cones. How many different combinations of one flavor and one kind of cone are there?

Basic problem types

Multiplication is: So many times larger (or smaller)…• Multiplicative comparison problems: My mom is 6 times older

than I am. I am 7 years old. How old is my mom?

Foundational teaching

When students are given these kinds of problems and asked to work them in any way they can (and given manipulatives), they generally use these kinds of strategies:• Skip counting with smaller numbers, then counting on• Strategies for deriving facts, such as knowing doubles, “double -

doubles” for multiplying by 4, “times ten then 1/2” for multiplying by 5, etc.

• Some combinations are easier to remember, such as the square numbers. A strategy based on the square numbers is the “off 1” strategy: the product of the two numbers “around” another (6 and 4 are “around” 5) is one less than the square of the middle number.

• Knowing the fact families for products of numbers results in knowing the division facts.

Learning Progression

Are there more steps to this?Or are some out of order?

Or does the order matter?Or are some inconsequential?

for multi-digit numbers

Area Models

Area Models

Area Models

• National Library of Virtual Manipulatives• http://nlvm.usu.edu/ 3-5 grade Number and Operations

Rectangle Multiplication

C-R-A for multiplication

Objects Pictures Symbols

How do you connect this?

• How would you connect the learning progressions (content) with the instructional strategies (practices)? Look back through the powerpoint slides at the content, look at the instructional strategies on the walls.

• What changes should we see in our instruction?

Formative feedback

• Looking at the results of the little assessment, can you spot areas that need attention? Whole class or small group?

Intervention resources

Multiplying decimals

National Library of Virtual Manipulativeshttp://nlvm.usu.edu/ 3-5 Number and Operations – Base Blocks Decimals3-5 Number and Operations – Percent Grids

How would you modify your C-R-A?

Formative feedback

Multiplying fractions

National Library of Virtual Manipulativeshttp://nlvm.usu.edu/ 3-5 grade Number and Operations Fractions – Rectangle Multiplication

How would you modify your C-R-A?

Basic division concepts

Division is 1. partitive (fair shares) (3.OA.2)

Megan has 15 cookies. She put the cookies into 5 bags with the same number of cookies in each bag. How many cookies are in each bag?

2. measurement (goes into)Megan has 15 cookies. She puts 3 cookies in each bag. How many bags can she fill?

Remainders…

• The remainder is simply left over and not taken into account (ignored)

• The remainder means an extra is needed• The remainder is the answer to the problem• The answer includes a fractional part• Nat’l Library of Virtual Manip’s: Rectangle division

C-R-A for division

C-R-A for division

C-R-A for division

How can you show this procedure symbolically?

Formative feedback

• What did you learn from the assessments?

Something to try…

• Choose something from today’s session (or a couple things) to try before the next session. Come back with examples of students’ work from what you tried.

• Next session Dec. 5, here.